G.587 side hexagon decagon — Code

XIII.9

If the side of the hexagon and that of the decagon inscribed in the same circle be added together, the whole segment has been cut in extreme and mean ratio, and its greater segment is the side of the hexagon.

"""Programmatic construction for Book XIII, Proposition 9 (XIII.9).

        Circle ABC with centre E, diameter BA, side of decagon BC, and side of
        hexagon CD in a straight line with BC.
        """

        from __future__ import annotations

        import math
        from geometor.model import Model


        def construct_g1() -> Model:
            """Construct the figure 0459.g1 using geometor.model."""
            model = Model("XIII.9")

            cx, cy, r = 114.0, -96.0, 82.0
            E = model.set_point(cx, cy, classes=["given"], ID="E")
            B = model.set_point(cx - r, cy, classes=["given"], ID="B")
            A = model.set_point(cx + r, cy, classes=["given"], ID="A")

            # Circle ABC
            model.construct_circle(E, A)

            # C is 36 degrees below B on the circle
            c_rad = math.radians(-144.0)
            C = model.set_point(
                round(cx + r * math.cos(c_rad), 1),
                round(cy + r * math.sin(c_rad), 1),
                classes=["given"],
                label="C",
            )

            # CD is in a straight line with BC and has length r (side of hexagon)
            dir_rad = math.radians(-72.0)
            D = model.set_point(
                round(47.7 + r * math.cos(dir_rad), 1),
                round(-144.2 + r * math.sin(dir_rad), 1),
                classes=["constructed"],
                label="D",
            )

            # F is intersection of ED with the circle (at -108 degrees)
            f_rad = math.radians(-108.0)
            F = model.set_point(
                round(cx + r * math.cos(f_rad), 1),
                round(cy + r * math.sin(f_rad), 1),
                classes=["constructed"],
                label="F",
            )

            # Triangles
            model.set_polygon([E, B, C])
            model.set_polygon([E, B, D])
            model.set_polygon([E, C, D])

            # Segments and collinear sub-segments
            for p1, p2 in [
                (B, A), (B, E), (E, A),
                (B, D), (B, C), (C, D),
                (E, C), (E, D), (E, F), (F, D),
            ]:
                model.set_segment(p1, p2)

            # Sections
            model.set_section([B, E, A])
            model.set_section([B, C, D])
            model.set_section([E, F, D])

            # Angles (wedges)
            model.set_wedge(E, A, A, C)
            model.set_wedge(E, C, C, B)
            model.set_wedge(B, E, E, C)
            model.set_wedge(C, E, E, B)
            model.set_wedge(E, C, C, D)
            model.set_wedge(D, C, C, E)
            model.set_wedge(E, B, B, D)

            return model


        if __name__ == "__main__":
            construct_g1()